Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary [Formula: see text]-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian [Formula: see text]-spaces, which (as we show) unfortunately fails to mirror the situation where more than one [Formula: see text]-module “quantizes” a given Hamiltonian [Formula: see text]-space. This paper offers evidence that the situation is remedied by working in the category of prequantum [Formula: see text]-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces and establish Frobenius reciprocity as well as the “induction in stages” property.